1. Field of the Invention
The present invention relates generally to fuel assemblies for a nuclear reactor and, more particularly, to a method of searching for and optimizing fuel placement and burnable absorber numbers and placement in a nuclear reactor core. The invention also relates to an algorithm for optimizing core loading patterns.
2. Background Information
The core of a modern commercial nuclear power reactor is formed by numerous elongated, rectangular fuel assemblies arranged in a cylindrical vessel. The fuel assemblies are arranged in accordance with a loading pattern intended to meet certain engineering requirements, such as the distribution of power, including limits on power peaks within the core. Other considerations include the maximization of the fuel cycle length, or the time required between refuelings. The loading configuration and plan of replacement and arrangement of fuel during the life of the reactor is known as in-core fuel management, and is a major reactor design consideration.
It is common to combine fuel assemblies from previous fuel cycles with new fuel. Accordingly, a typical inventory of fuel assemblies might comprise one-third new, or feed fuel assemblies, about one-third once burned fuel assemblies and about one-third twice burned fuel assemblies. While fuel assemblies within these broad categories will have similar gross characteristics, there are differences that must be taken into account even among fuel assemblies with a common history. Further complicating in-core fuel management, is the fact that the feed fuel assemblies typically include the addition of a number of different arrangements of burnable absorbers in order to moderate their reactivity.
Traditionally, one of two techniques has been employed to establish a suitable fuel assembly loading pattern for a reactor core. In one technique, a sequence of random shuffles of fuel assemblies is used to find a loading pattern that meets specified engineering requirements. In the other technique, expert rules, developed through years of experience, are used to establish a suitable loading pattern. However, neither of these techniques assures the establishment of an optimal loading pattern.
More recently, a variety of analytical techniques, including algorithms for use individually or in combination with computer programs or solvers, have been developed and employed in order to search for and optimize core loading patterns based upon principles of optimization.
One such analytical approach was suggested in a paper entitled “Mixed Integer Programming for Pressurized Water Reactor Fuel-Loading-Pattern Optimization”, by Taek Kyum Kim and Chang Hyo Kim, published in Nuclear Science and Engineering: 127, 346-357 (1997). This technique begins by assuming an arbitrary initial reference loading pattern of the individual fuel assemblies. The number of possible loading patterns that could be generated by shuffling all of these individual fuel assemblies is formidable. Therefore, it is not practical to individually evaluate each possible loading pattern discretely. In order to reduce the processing required, the problem is expressed as a continuous function that can then be differentiated to obtain an optimum solution. This is accomplished by linearizing the relationship between the physical distribution of the fuel assemblies and the pertinent engineering requirements using a generalized perturbation technique. Mixed integer linear programming incorporating branching and bounding is then implemented to determine the optimum loading pattern. As the linearization to individual fuel assemblies can introduce significant error, there is no assurance that the “optimal” loading pattern identified is in fact optimal. Accordingly, the process is repeated using the solution generated by the previous integration. However, there is no guarantee that subsequent solutions will converge or even that a better loading pattern will be found. The process is also heavily dependent on the initial reference loading pattern used.
Another method of establishing a fuel assembly loading pattern is described in the paper entitled “Integer Permutation Programming in a Loading Pattern Optimization Code SUPERLPOS Used at SNERDI” presented by Shengyi Si, at the PHYSOR 2002 Conference at Seoul, Korea on Oct. 7-10, 2002. This technique involves a two part process. First, the fuel assemblies for a load are grouped into batches based upon initial enrichment and depletion history. For instance, all of the feed fuel assemblies are considered one batch, the once burned fuel assemblies form a second batch, and the twice burned fuel assemblies make up the third batch. Initially, all of the fuel assemblies in each batch are considered to be identical. An enumeration program sequentially generates batch loading patterns one by one by shuffling the batches of fuel assemblies. For each enumerated batch loading pattern the reactivities of the individual fuel assemblies are utilized in the linearization of the effect of fuel assembly position on the specified engineering requirements. Again, mixed integer linear programming incorporating branching and bounding is applied to identify an optimal loading pattern of individual fuel assemblies for the selected batch pattern. The second part of the technique is similar to the aforementioned technique, except that the linearization using the same generalized perturbation technique has smaller errors due to the smaller perturbations required when shuffling individual assemblies inside a batch. In accordance with this technique, different batch loading patterns are serially enumerated and processed to find the optimal daughter loading pattern for each batch loading pattern. A final loading pattern is then manually selected from all of the optimal daughter loading patterns that a user cares to generate.
While this latter technique is an improvement, it still has limitations. First, it does not identify the size of the problem, in that batch loading patterns are serially enumerated and processed. Additionally, by going directly from the batch groupings to the individual fuel assemblies in one step, there can be wasted time processing patterns that have a dead end, where none of the daughter loading patterns meet the specified engineering requirements. Second, jumping to the individual fuel assemblies in one step, even though the differences are not large between fuel assemblies in the same batch, nonetheless produces sizeable errors in the linearization.
A still further method of finding fuel assembly loading patterns involves performing an exhaustive and deterministic search for all patterns satisfying specified engineering requirements. This method is disclosed in U.S. patent application Ser. No. 10/374,691 entitled: “METHOD OF ESTABLISHING A NUCLEAR REACTOR CORE FUEL ASSEMBLY LOADING PATTERN.” First, all of the loading patterns possible from an inventory of fuel assemblies grouped into batches of like coarse levels of reactivity, are enumerated, taking into account selected core load position constraints. The resulting parent loading patterns are then all analyzed for compliance with certain engineering requirements, such as peak power distribution. Next, the parent loading patterns that satisfy the specified engineering requirements are each processed further by refining the original batches assigned by coarse levels of reactivity into a larger number of smaller batches according to finer levels of reactivity. If any of the resulting daughter patterns meets the specified engineering requirements, an optimal daughter pattern is selected using the well-established branch and bound mixed integer linear programming method, in which a more accurate direct perturbation technique is used instead of the generalized perturbation technique. The parent patterns are then replaced by the optimal daughter patterns and the process is repeated in order to generate patterns from even finer batches of fuel assemblies. This scheme of batch refinement and branch and bound mixed integer linear programming is repeated until the batches comprise individual fuel assemblies. Additionally, before the batch sizes are reduced to single fuel assemblies, arrangements of boron absorbers are assigned to the feed fuel assemblies. Typically there will be multiple optimal fuel loading patterns, satisfying the requisite engineering requirements, at the individual fuel assembly level. Therefore, selection of the pattern to be used can be based on criteria established by the customer, such as the optimal pattern that produces the longest fuel cycle. This final step can be performed manually or implemented automatically.
The foregoing method, although constituting a refinement in the art of core loading pattern searching and optimization, still suffers from a unique set of disadvantages. For example, it requires repeated generation of constraint matrices which are produced by perturbing the reactivity of individual assembly locations one at a time, and then calculating the effects of these perturbations for the cycle. It also requires the repeated running of a commercially available solver, such as a mixed integer program (MIP) solver. Despite significant advances in computer technology, generation of such matrices can take a prohibitively long time to perform, in some cases up to 24 hours or more. Then additional time, again potentially lasting hours in duration, is spent running the MIP solver to find the optimum.
There is also no standard or uniform run time for these calculations, making them unpredictable. The method further requires the enumeration and processing of the batch loading patterns and, in order to control linearization errors, the optimization decisions must be decomposed into three stages. This decomposition separates the fine batch assignment optimization from the burnable absorber (BA) assignment optimization, from the final assembly assignment optimization. This process can prevent the method from finding the best optimum. Furthermore, linearizing away the “shuffling non-linearity” in the nuclear flux solutions is likely to result in larger errors than if it were not linearized away. The “shuffling non-linearity” is the non-linearity caused by the fact that a shuffle changes both the neutron flux at a core location as well as the fuel composition at that location. Changing the BA assignment in a location results in a similar non-linearity for the same reason.
As it is the neutron distribution that determines the various interaction rates and energy production rate within the core, it is desireable to provide a method of core loading pattern searching and optimization that represents the shuffling non-linearity as accurately and efficiently as possible, for example by representing it directly in the constraint matrix supplied to the solver (e.g., MIP solver). This would minimize the accumulation of errors in the optimization, while simultaneously streamlining the efficiency and, in particular, the timeliness of the method optimization.
There is room, therefore, for an improved method for establishing loading patterns for nuclear reactor cores and in algorithms for facilitating searching for and optimizing core loading patterns.